Escher's Square Limit is based on the same geometric scaffolding as Regelmatige vlakverdeling, Plate VI. Find four copies of the Plate VI geometry in the Square Limit geometry and draw a sketch to show where they are. Note that the Plate VI geometry is “house” shaped, and the Square Limit is a square, so draw a square and put (at least) four “houses” in it.

Sketch the underlying geometric scaffolding for Sketch #101 (Division). Use graph paper, and make a 45°-45°-90° triangle for each lizard. As a hint, start with a rectangle that is 16 squares wide by 8 squares high for the top row of four lizards.

What’s going on in Division? What is dividing, and is there any pattern to it?

On a fresh piece of graph paper, draw one small square. Draw another square next to it on the right, making a 2x1 rectangle. Now draw another square along the long edge of the 2x1 rectangle, making a 3x2 rectangle. Continue this process, spiraling outward, until you're out of room on the page:
Make a table showing the side lengths of each rectangle. Calculate the ratio of the long side to the short side for each rectangle.

In the previous problem, the rectangles changed shape less and less as the process continued (the ratios of long to short sides didn't change much). Suppose you want a rectangle that stays exactly the exactly the same shape when a square is attached:
File:Golden-rectangle.svg
This means the side ratios must be equal: . Solve this equation for . (Cross multiply and use the quadratic formula!)

Calculate the ratio of your height to the height of your navel. Do the same for four friends. Compare your results with the previous two problems.